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SPHERES OF PRESCRIBED MEAN CURVATURE IN S 3 MICHAEL T. ANDERSON
 

Summary: SPHERES OF PRESCRIBED MEAN CURVATURE IN S 3
MICHAEL T. ANDERSON
Abstract. We prove a ``semi­global'' result on the existence of conformal embeddings of S 2 into
S 3 of prescribed mean curvature.
1. Introduction
In this note, we discuss the existence of conformal embeddings of a two­sphere S 2 with prescribed
mean curvature into the round 3­sphere S 3 of radius 1.
To state the main result, let C 1 = C m-1,#
1 denote the space of C m-1,# functions H : S 2
# R
satisfying
0 < H < 1
where m # 3, # # (0, 1). Define an equivalence relation on C 1 by
(1.1) [H 1 ] = [H 2 ] # H 2 = H 1 + #,
where # = # c i x i is the restriction of a linear function to S 2
# R 3 , so that the gradients ## form
the space of conformal vector fields on S 2 . Let D 1 = C 1 / # be the quotient space.
Theorem 1.1. For any pointwise C m,# conformal class [#] of metrics on S 2 and for any equivalence
class [H] # D 1 , there exists a C m+1,# embedding of S 2 into S 3 with prescribed pointwise conformal
class [#] and prescribed mean curvature class [H]. Thus, there exists a C m+1,# smooth embedding

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics